ON THE RATE OF CONVERGENCE OF CERTAIN LINEAR POSITIVE OPERATORS

Abstract

In the present thesis, we study approximation properties of some well known operators and their q-analogues. We divide the thesis into seven chapters. The chapter 0 includes the literature survey, basic de nitions and some basic notations of approximation methods which will be used throughout the thesis. In the rst chapter, we obtain some approximation properties of Durrmeyer type modi cation of generalized Baskakov operators. We discuss some direct results in simultaneous approximation by these operators e.g. the pointwise convergence theorem, Voronovskaja-type theorem, an estimate of the error in terms of the modulus of continuity and the approximation of functions having derivatives of bounded variation for these operators. The second chapter deals with the q-analogue of Bernstein-Schurer-Stancu operators. The Voronovskaja type theorem, local and global direct results have been studied for these operators. In the third chapter, we propose the Stancu type generalization S( ; ) n;p (:; q; x) of the modi ed Schurer operators based on q integers. We obtain the local direct results and statistical convergence results for these operators. We also introduce the limit q-modi ed Schurer-Stancu operators and study their rate of convergence. In the fourth chapter, we propose the q-analogue of certain Baskakov-Durrmeyer type operators. First, we establish the recurrence relation for the moments of the operators by using the q-derivatives and then prove the basic convergence theorem. Next, some local direct results and weighted approximation properties for these operators are discussed. Lastly, we study the King type approach in order to obtain the better approximation for these operators. i ii In the fth chapter, we study some approximation properties of the Kantorovich type modi cation of q-Bernstein-Schurer operators. We obtain some local direct results for these operators, e.g. local approximation, rate of convergence by means of modulus of smoothness and Lipschitz type maximal function. Further, we study Voronovskaja type theorem and A-statistical convergence of these operators. Lastly, we construct the bivariate generalization of these operators and study some local direct results. In the last chapter, we obtain the upper bound, exact order of approximation and Voronovskaja type results with quantitative estimates for the complex q-modi ed Bernstein-Schurer operators (0 < q < 1) attached to analytic functions on compact disks. In this way, we show the overconvergence phenomenon for these operators, namely the extensions of approximation properties from the real intervals to compact disks in the complex plane.